Mine too will be a partial answer.

I'll include data that I can quickly summarize and provide any other useful insights (such as any formulae) I can think of. I'm not familiar with the website so I can't do a comparison / critique but plan to investigate it in the future.

Star Generation

Depending upon the response, I might put more of what I have up here. But this took me the last 4 hours to put together.

Assumptions

In my own notes, I do not identify the assumptions I made - I simply made them, move on, and forgot them. I'll try to document them as I go through my materials so some of my answers don't seem so mysteriously disconnected from reality.

• A random distribution of stars - the reality is the type of star you see will depend heavily upon where you look. My investigation assumed a jump point interstellar drive system and that the connections were random (it made life easier).
• I've extended the spectral classifications to make my life easier. Conventional spectral classes include I, II, III, IV, V, VI. I've added VI, VII, VIII, IX, X, and XI as shown in the table below.

$$\begin{array}{|c|c|c|} \hline \text{Luminosity Class} & \text{Description} & \text{Origin of Class} \\ \hline \text{I} & \text{Supergiant} & \text{Conventional Classification} \\ \hline \text{II} & \text{Brightgiant} & \text{Conventional Classification} \\ \hline \text{III} & \text{Giant} & \text{Conventional Classification} \\ \hline \text{IV} & \text{Subgiant} & \text{Conventional Classification} \\ \hline \text{V} & \text{Dwarf} & \text{Conventional Classification} \\ \hline \text{VI} & \text{Subdwarf} & \text{Conventional Classification} \\ \hline \text{VII} & \text{White Dwarf (stellar remnant)} & \text{Conventional Classification} \\ \hline \text{VIII} & \text{Neutron Star} & \text{My extension} \\ \hline \text{IX} & \text{Blackhole} & \text{My extension} \\ \hline \text{X} & \text{Nova, Supernova, other} & \text{My extension} \\ \hline \text{XI} & \text{Hypernova} & \text{My extension} \\ \hline \end{array}$$

Frequency of stellar objects

I downloaded and analyzed several star catalogs and determined star probabilities using that information. My analysis revealed the patterns seen below. Be aware that the numbers for the stellar objects with a Luminosity Class above VI (subdwarf) is estimated by me and probably bears little resemblance to reality.

I use type and class to categorized stars and also use the H-R (Hertzsprung-Russell) diagram. The table provided below was developed primarily by

$$\begin{array}{|c|c|c|c|c|} \hline \text{Lum Class} & \text{Spect Class} & \text{Description} & \text{Probability} & \text{Cumulative} \\ \hline \text{I} & \text{M+} & \text{Supergiant} & \text{2.45%} & \text{2.45%} \\ \hline \text{II} & \text{M+} & \text{Brightgiant} & \text{0.97%} & \text{3.42%} \\ \hline \text{III} & \text{M+} & \text{Giant} & \text{10.14%} & \text{13.56%} \\ \hline \text{IV} & \text{M+} & \text{Subgiant} & \text{4.58%} & \text{18.14%} \\ \hline \text{V} & \text{O} & \text{Ultraviolet Dwarf} & \text{0.16%} & \text{18.30%} \\ \hline \text{V} & \text{B} & \text{Blue Dwarf} & \text{1.84%} & \text{20.14%} \\ \hline \text{V} & \text{A} & \text{Blue-white Dwarf} & \text{1.70%} & \text{21.84%} \\ \hline \text{V} & \text{F} & \text{White Dwarf} & \text{2.29%} & \text{24.13%} \\ \hline \text{V} & \text{G} & \text{Yellow Dwarf} & \text{2.90%} & \text{27.03%} \\ \hline \text{V} & \text{K} & \text{Orange Dwarf} & \text{10.69%} & \text{37.72%} \\ \hline \text{V} & \text{M} & \text{Red Dwarf} & \text{53.01%} & \text{90.73%} \\ \hline \text{V} & \text{L} & \text{Infrared Dwarf} & \text{6.20%} & \text{96.93%} \\ \hline \text{VI} & \text{M - G} & \text{Subdwarf} & \text{0.24%} & \text{97.17%} \\ \hline \text{VII} & \text{G+} & \text{White Dwarf(stellar remnant)} & \text{2.40%} & \text{99.57%} \\ \hline \text{VIII} & \text{F+} & \text{Neutron star} & \text{0.31%} & \text{99.88%} \\ \hline \text{IX} & \text{N/A} & \text{Blackhole} & \text{0.10%} & \text{99.98%} \\ \hline \text{X} & \text{N/A} & \text{Supernova, Nova, other} & \text{0.019%} & \text{99.999%} \\ \hline \text{XI} & \text{N/A} & \text{Hypernova} & \text{0.001%} & \text{100%} \\ \hline \end{array}$$

• M+ - Object can have a spectral class of M - O
• K+ - Object can have a spectral class of K - O
• G+ - Object can have a spectral class of G - O
• F+ - Object can have a spectral class of F - O
• M - G - Object can have a spectral class of M - G

Star Generation

To generate your very own star, you may use the following steps:

1. Generate a random number ($N_{random}$) between 0 & 1.
2. Apply to the table above.
3. If the number falls outside the range of Luminosity Class V - main sequence dwarf (e.g. $0.1814 > N_{random} > 0.7900$), you are done.
4. Otherwise, $N = N_{random} - 0.1814$
5. $M_{star} = \frac{C_1}{C_2 + C_3 \cdot N}$ in Solar Masses
6. $D_{star} = M_{star}^{2/3}$ (assumes main sequence stars have roughly the same density) in solar diameters
7. $T_{star} = 380 + 5500 \cdot \sqrt{M_{star}}$ in K
8. $L_{star} = \frac{4 \cdot \pi \cdot \left(D_{star} \cdot 7 \cdot 10^8 \right)^2 \cdot T_{star}^4 \cdot \sigma}{L_{\odot}}$ in Sols
9. Maximum age $T_{star} = \frac{1.15 \cdot M_{star}}{L_{star}} \cdot 10^{10}$ in years
10. The middle of the Goldilocks zone $r_{planet} = \sqrt{L_{star}}$ in AU

The constants

• $C_1 = 0.0171$
• $C_2 = 0.000143$
• $C_3 = 0.271$
• $\sigma = 5.67 \cdot 10^{-8}$ - Boltzmann's Constant

Wikipedia has a more accurate Luminosity calculation but it is a multiple curve fit solution and it only gives significantly different answers at the extreme high and low end of star masses.

This is a great thing to calculate and store in a spreadsheet, which is what I do.